The Spurs' victory, combined with Houston's loss to Dallas, made San Antonio champion of the Southwest Division and the No. 3 seed in the Western Conference playoffs.
San Antonio will take on sixth-seeded Dallas in the opening round. New Orleans slipped behind Dallas into the No. 7 seed on the final night of the regular season and will face Denver in the first round.
James Posey gave New Orleans a three-point lead with 8 seconds to go in the fourth quarter, but Finley stunned the Hornets with his 3-pointer just before the buzzer that forced the extra period.
Tony Parker and Tim Duncan then combined to score 11 points in the overtime to give San Antonio a regular-season record of 53-28. Houston went into the final night of the season tied with the Spurs for the top spot in the division, but held the tiebreak advantage.
Houston, however, slipped to 52-29 with its loss at Dallas.
Parker led the Spurs with 29 points, Duncan had 20 and Finley 17. David West paced the Hornets with 34 points and Chris Paul had 26.